Links to interesting surveys in Geometry and Topology (not mine):
Heegard-Floer homology.
What is a pseudoholomorphic curve?
What is a tropical curve?
What is an open book?
What is a Lefschetz pencil?
What is an amoeba?
Some AIM K3 lists of open problems in mathematics:
Syzygies and Mirror Symmetry.
Mahler’s conjecture and duality in convex geometry.
Fourier analytic methods in convex geometry.
Sections of Convex Bodies.
Open problems and questions about geodesics.
Amoebas and Tropical Geometry.
Holomorphic curves in Contact Geometry.
Variational methods in Celestial Mechanics.
Gromov-Witten invariants and Number Theory.
Higher dimensional Contact Topology.
Quantum invariants and low dimensional topology.
Contact Topology in higher dimensions.